Session RZ: Waves III

On weakly nonlinear gravity-capillary solitary waves

As a weakly nonlinear model equations system for gravity-capillary solitary waves on the surface of a potential flow, a cubic-order truncation model is presented, which is derived from the Taylor series expansion of the Dirichlet-Neumann operator (DNO) for the free boundary conditions of the Euler equations in terms of Zakharov's canonical variables. In deep water, the cubic-order truncation model allows gravity-capillary solitary wavepackets in the weakly nonlinear and narrow bandwidth regime where the classical nonlinear Schr\"{o}dinger (NLS) equation governs. Since this model is consistent to the original full Euler equations in the order of nonlinearity up to the third order, the properties of the gravity-capillary solitary waves of this model precisely agree with the counterparts of the Euler equations. From this cubic order truncation model, the leading-order initial long-wave transverse instability growth rate of the gravity-capillary solitary waves is estimated to be identical, in the weakly nonlinear limit, to the earlier result by Kim and Akylas (J. Eng. Math. 58:167-175, 2007), through an equivalent perturbation procedure. Based on these analytical and numerical observations, the cubic-order truncation model equations system is regarded as the optimal reduced model for the dynamics of weakly nonlinear gravity-capillary solitary waves.   

Radiation and diffraction of surface waves by an oscillating water column

An oscillating water column (OWC) modelling a wave energy converter is studied in a small scale laboratory experiment. The system consists of a cylindrical duct partially submerged in a wave tank. The water surface elevation inside the duct oscillates in response to the forcing imposed by an external wave field. The oscillation amplitude inside the duct is maximized when a resonance condition is attained. From a wave energy conversion point of view, the physical phenomena limiting the amplitude of the oscillations can be viewed as energy losses, and they can be mainly classified in friction losses inside the duct, losses due to vortex formation around the duct mouth, and losses due to radiation of surface waves produced by the OWC. Here we study the wave field around the half-submerged duct using a 2D profilometry technique, which permits an accurate measurement of the free surface height over a large field of view. We characterize the radiation and diffraction wave fields as a function of the detuning of the monochromatic forcing wave to the resonance frequency and give an estimate of the energy losses due to radiated waves.   

Initial development of a corner wave

We have studied the flow downstream the corner of a partially submerged vertical plate using a combination of experimental, numerical and analytical tools. In this flow configuration, a steady wave remains attached to the corner of the plate. Both the amplitude and slope of the wave front increase with the downstream distance until the wave breaks resulting in either a spilling or a plunging breaker. This simple laboratory set-up can be used to gain a better understanding on how waves break in other configurations of interest in oceanography or naval hydrodynamics. In particular, the effect of two dimensionless parameters on the breaking process is explored: (1) a Froude number based on the height of the free surface above the plate corner and (2) the dimensionless curvature of this corner. It will be shown that, properly re-scaling the trajectory of the corner wave front with the above mentioned parameters, it follows a universal curve during the first instants of the formation process.   

Wave Generation Experiments Using a Cycloidal Turbine

We investigate the wave generation performance of a cycloidal turbine for the purpose of converting wave energy to shaft power. Cycloidal turbines consist of one or more hydrofoils that rotate around a central shaft and can be pitched during rotation. In the present investigation, a two-dimensional wave channel of 45cm width, 4.5m length and a water depth of 30 cm is used. It features a flap wave maker at one end, and a beach at the other end. A two blade Cycloidal turbine model is placed in the center of the wave channel and the generated waves in both up-wave and down-wave directions are measured using wave gauges. We compare the results to inviscid potential flow simulations that show negligible waves traveling up-wave, and a single harmonic wave traveling down-wave making the Cycloidal turbine an ideal wave energy converter if synchronized to the incoming wave.   

Simulation of turbulence interacting with free surface and wave

Direct numerical simulation is performed for homogeneous turbulence interacting with deformable free surfaces and progressive waves, respectively. For the free surface case, various Froude and Weber numbers are considered. Surface manifestations of the underlying turbulence in the forms of propagating waves and surface roughness are elucidated. Effects of splats and anti-splats on turbulence kinetic energy budget are quantified. For the progressive wave case, effects of wave strain field and Stokes drift are examined. It is found that turbulent Reynolds stress is strongly dependent on the wave phase. Wave normal production, pressure-strain correlation, and pressure transport are essential in the Reynolds stress budget. Vortices are turned, stretched, and compressed periodically by the wave strain field, leading to their wave-phase dependent distribution. Lagriangian average shows that both the Stokes drift and the high correlation between wave strain field and turbulence contribute to the turning of vertical vorticity into streamwise direction.   

Lattice Boltzmann Simulations for Wave Propagation

In the past two decades, the lattice Boltzmann method(LBM) has attracted much attention as an alternative approach to the traditional methods in computational fluid dynamics. It possesses certain advantages in solving many problems over conventional methods. Here, we focus on the lattice Boltzmann model for wave equations. Firstly, in order to obtain wave equations with higher-order accuracy of truncation errors, we removed the second-order dissipation term and the third-order dispersion term by employing the moments up to fourth order in the lattice Boltzmann models with the classical Chapman-Enskog expansion. The time reversibility seems due to the accurate mimicking of the wave equations up to 4$^{th}$ order, that is the absences of the second-order dissipation term and the third-order dispersion term. Secondly, the numerical verification for the model have been carried out, some classical examples are simulated, including wave interference, diffraction, and wave passing through a convex lens. The numerical results demonstrate that the model can be used efficiently to simulate wave propagations in various situations. Acknowledgement: This research is supported in part by Ministry of Education in China via project IRT0844 and NSFC project 10625210 and Shanghai Sci and Tech. Com. Project 08ZZ43   

Gravity currents in two-layer stratified media

An investigation of gravity currents propagating through a two-layer stratified ambient of finite vertical extent is presented. Our theoretical discussion considers slumping, supercritical gravity currents, i.e.~those that generate an interfacial disturbance whose speed of propagation matches the front speed. In contrast to previous studies, we parameterize the amplitude of the interfacial disturbance; the accuracy of this approach is confirmed by comparison against experimental and numerical data. Measured front speeds show positive agreement with analogue model predictions, which remain strictly single-valued. The front speed is essentially independent of the interfacial thickness, $\delta$, even in the limiting case where $\delta=H$ so that the environment is comprised of a uniformly stratified ambient with no readily discernible upper or lower ambient layer. Our experiments also consider the horizontal distance, $X$, at which the front begins to decelerate revealing a non-monotonic dependence on the ambient interface height.